Contrasting behaviour of two spherically symmetric perfect fluids near a weak null singularity in a spherically symmetric black hole

Abstract

In this work we contrast the behaviour of two spherically symmetric matter models in a class of spherically symmetric spacetimes which feature a weak null singularity. This class in particular contains spherically symmetric perturbations of subextremal Reissner-Nordström under the Einstein--Maxwell--scalar field system, a system for which a $C^2$ formulation of the strong cosmic censorship conjecture was proved by Luk-Oh, arXiv:1702.05715 and Dafermos, arXiv:1201.1797. Firstly, we consider the Cauchy problem of spherically symmetric dust falling into the weak null singularity (WNS) where the initial dust velocity is normal to a smooth spacelike curve with certain properties. We prove that the flow of the dust velocity does not experience any shell-crossing before or at the singularity, the velocity vector remains timelike, and that the dust energy density remains bounded as matter approaches the singularity. Secondly, we consider the characteristic initial value problem for stiff perfect fluid falling into the WNS. By relating the stiff fluid velocity and energy density to a scalar field satisfying the homogeneous linear wave equation, we prove that this energy density becomes infinite as we approach the weak null singularity. Furthermore, we show that the ingoing component of the stiff fluid velocity blows up while the outgoing component approaches zero at the singularity. Therefore the velocity vector approaches an ingoing null vector tangent to the singular hypersurface.

Figures (18)

• Figure 1.1: Left: Illustration of the statement Theorem 1. The flow lines of dust, shown in blue, are initially normal to the spacelike curve $\lambda$ and extend to the singularity as timelike curves. The energy density of dust remains bounded. Right: Illustration of the statement of Theorem 2. The flow lines of the stiff fluid, in blue, emerge as timelike curves from the bifurcate null hypersurface. However, they approach ingoing null curves in the vicinity of the singularity. The energy density of the stiff fluid blows up.
• Figure 2.1: Penrose diagram illustrating the spacetime $\mathcal{R}$, shown in yellow. Note that $\mathcal{R}$ coincides with the domain of dependence of the spacelike hypersurface $\Sigma =\{uv=u_{s}v_{\kappa}\}$.
• Figure 3.1: Admissible curve $\lambda$ with future-directed unit normal $\nu$ and tangent $\lambda'$.
• Figure 3.2: Illustrating the geodesic variation based on the admissible curve $\lambda$. Although in this picture there is no shell-crossing between the geodesics, this is a nontrivial result we are yet to prove. Key features in the illustration are that $\alpha\to \overline{\Gamma}(0,\alpha)$ is the curve $\lambda$, while by definition of the proper time function $T(\alpha)$, $\alpha\mapsto\overline{\Gamma}(\alpha,T(\alpha))$ maps to the weak null singularity. We will see that the restriction $\Gamma$ to $D^0$ maps away from the singularity i.e. away from the boundary of $\overline{\mathcal{Q_R}}$.
• Figure 4.1: The Jacobi field along a future-directed inextendible radial timelike geodesic based on an admissible curve $\lambda$.

Theorems & Definitions (100)

• Theorem 1.1
• Theorem 1.2
• Remark
• Remark
• Remark
• Definition 3.1
• Definition 3.2
• Definition 3.3
• Theorem 3.4
• Definition 3.5